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A295366
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - b(n-3), where a(0) = 1, a(1) = 2, a[2] = 3, b(0) = 4, b(1) = 5, b(2) = 6, and (a(n)) and (b(n)) are increasing complementary sequences.
2
1, 2, 3, 12, 23, 44, 77, 132, 221, 367, 604, 987, 1608, 2613, 4240, 6873, 11134, 18029, 29186, 47240, 76453, 123720, 200201, 323950, 524181, 848162, 1372375, 2220570, 3592979, 5813584, 9406599, 15220220, 24626857
OFFSET
0,2
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295357 for a guide to related sequences.
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
FORMULA
a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).
EXAMPLE
a(0) = 1, a(1) = 2, a[2] = 3, b(0) = 4, b(1) = 5, b(2) = 6, so that
b(2) = 7 (least "new number")
a(3) = a(2) + a(1) + b(2) + b (1) - b(0) = 12
Complement: (b(n)) = (4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, ...)
MATHEMATICA
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; a[2] = 3; b[0] = 4; b[1] = 5; b[2]=6;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] - b[n - 3];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
z = 32; u = Table[a[n], {n, 0, z}] (* A295366 *)
v = Table[b[n], {n, 0, 10}] (* complement *)
CROSSREFS
Sequence in context: A122604 A282234 A024780 * A323120 A323121 A323122
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 22 2017
STATUS
approved